Darboux integrable system with a triple point and pseudo-abelian integrals

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Darboux integrable system with a triple point and pseudo-abelian integrals

Aymen Braghtha

To cite this version:

Aymen Braghtha. Darboux integrable system with a triple point and pseudo-abelian integrals . 2016.

�hal-01388054v3�

DARBOUX INTEGRABLE SYSTEM WITH A TRIPLE POINT AND PSEUDO-ABELIAN INTEGRALS

AYMEN BRAGHTHA

ABSTRACT. We study pseudo-abelian integrals associated with polynomial perturbations of Dar- boux integrable system with a triple point. Under some assumptions we prove the local boundedness of the number of their zeros. Assuming that this is the only non-genericity, we prove that the num- ber of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for nearby Darboux integrable foliations.

1. INTRODUCTION

Abelian integrals are integralsI(h) = R

γ(h)ηof polynomial or rational one-forms along cycles γ(h)∈H₁(H⁻¹(h)), H ∈C[x, y]. Abelian integrals appear as principal part of Poincar´e displace- ment function of the perturbationdH+εηalongγ(h). Their zeros are related to limit cycles born in the perturbation. In [11, 7], Varchenko and Khovanskiii prove the following result

Theorem 1.1. There exist a uniform bound, depending only on the degree of H and η, for the number of real zeros of Abelian integrals.

Arnold posed with insistence the analogous problem for more general polynomial perturbation of integrable systems. In particular, for perturbation of systems having a Darboux first integrals

H = Q

P_i^αi, P_i ∈ C[x, y]. Then instead of Abelian integrals, one encounters pseudo-Abelian integrals. Pseudo-Abelian integrals naturally appear in generalizations to classes bigger than the Hamiltonian class. The simplest case is the case of Darboux integrable planar foliations F = {M^dH_H = 0}, where H = Q

P_i^αi is a product of real powers of bivariate polynomials and M = QP_iis the so-called integrating factor, so the formω=M^dH_H is a polynomial one-form. Poincar´e- Pontryagin criterium claims that in the regular part the limit cycles of the perturbed foliations F ={ω+η= 0}can be born only from those cyclesδof foliationFfor which the integralR

δ η M

vanishes. This integral is called the pseudo-Abelian integral, and currently the main open question is the existence of a uniform bound on the number of its zeros.

2. FORMULATION AND MAIN RESULT

This paper is a part of program of Bobie´nski, Mardeˇsi´c and Novikov to extend the Varchenko- Khovanskii theorem from Abelian integrals to pseudo-Abelian integrals. After studying the generic cases [2, 9] and some nongeneric cases [1, 3, 4], in this work we study a nongeneric case. Let

Date: December 27, 2017.

2010Mathematics Subject Classification. 34C07, 34C08.

Key words and phrases. Abelian integrals, Limit cycles, Darboux integrability.

1

F ={M^dH_H = 0}denote a foliation with a triple point (assume it to be at the origin), where H =P₀

k

Y

i=1

P_iⁱ, M =P₀

k

Y

i=1

P_i, P_i ∈R[x, y], , _i >0,

so that the level curves{P₀ = 0},{P₁ = 0}and{P₂ = 0}intersect transversally two by two at the origin which is the only triple point. LetF_λ = {M_λ^dH_H^λ

λ = 0}be a foliation unfolding F, M_λ an integrating factor, where

H_λ =P_λ

k

Y

i=1

P_iⁱ, M_λ =P_λ

k

Y

i=1

P_i. (1)

Generically, a triple point unfolds into three saddles-type singular pointsp^λ₀, p^λ₁ andp^λ₂ correspond to the transversal intersections of level curves{P1 = 0}and{Pλ = 0},{P1 = 0}and{P2 = 0}, and{P2 = 0}and {Pλ = 0}. Here also appears a centerp^c_λ in the triangular region bounded by these levels curves- see Figure 1.

P_i= 0 {Pλ= 0}

P_j= 0 P₁= 0

P2= 0

𝑝₁^𝜆

𝑝2𝜆

𝑝₀^𝜆 𝑝_𝜆^𝑐

γ(λ, h)

FIGURE 1. The phase portrait ofH_λ

2

Assume that the system MλdHλ

Hλ has a family of cycles γ(λ, h) ⊂ H_λ⁻¹(h) filling a connected component ofR²\ {PλP1· · ·Pk = 0}, which we denote byD. The period annulusDis bounded by{P_λ = 0},{P₁ = 0},{P₂ = 0}and some separatrices{P_i = 0}, i= 3,· · · , k.

Consider a polynomial perturbation of the systemM_λ^dH_H^λ

λ. M_λdH_λ

H_λ +κη, η=Rdx+Sdy, R, S ∈R[x, y], κ >0.

The linear part in perturbation parameterκof the Poincar´e first return map is given by the pseudo- Abelian integral

I(λ, h) = Z

γ(λ,h)

η

M_λ. (2)

We prove the existence of a local bound for the number of zeros of pseudo-Abelian integrals.

LetΩ^k(λ, n₀, n₁,· · · , n_k;n)be the following, finite dimensional space of polynomial system with Darboux type first integral:

Ω^k(λ, n₀, n₁,· · · , n_k;n) := {M_λdH_λ

H_λ +κη:H_λ =P_λ

k

Y

i=1

P_iⁱ,

degP_λ ≤n₀,degP_i ≤n_i,deg(R, S)≤n}, where M_λ = P_λQk

i=1P_i. The parameters of the space Ω^k(λ, n₀, n₁,· · · , n_k;n) are positive ex- ponents(, ₁,· · · , _k), coefficients of polynomialsP_λ, P_iand coefficients of the polynomials per- turbation(R, S). We distinguish an open subsetΩ^k₀(λ, n₀, n₁,· · · , n_k;n)defined by the condition that the unperturbed system (withκ = 0) admits a period annulusDand the following genericity assumptions are satisfied

• A₁. The Darboux first integralH_λ is regular at infinity.

• A₂. ^∂P_∂λ^λ|_(0,0,0) 6= 0.

• A₃. The level curves{P_i = 0}, i= 3,· · ·, kare smooth and intersect transversally two by two. Morever, there are not triple intersection points, fori= 3, . . . , k.

• A₄. ηvanishes to the order≥4at(x, y) = (0,0)(technical condition)

Theorem 2.1. LetΩ₀ ∈ Ω^k₀(λ, n₀, n₁,· · · , n_k;n). Assume that the pseudo-Abelian integral I_Ω0 associated to Ω₀ in (2) is not identically zero. Consider the pseudo-Abelian integrals associated to Ω ∈ Ω^k₀(λ, n₀, n₁,· · · , n_k;n) in a neighborhood of Ω₀. Then, there exists an upper bound Z(Ω0;k, n0,· · · , nk;n)for the number of zeros of the pseudo-Abelian integrals associated toΩ∈ Ω^k₀(λ, n0, n1,· · · , nk;n)close toΩ0.

The main problem is that the Darboux integrable systemsM_λ^dH_H^λ

λ have a small nest of cycles which shrinks to the origine(0,0)asλ→0, i.e. the centerp^λ_c generates a possible ramification points of I(λ, h)located on a circle whose radius is of order|λ|⁺¹⁺². Thus, we can no directly repeat the argument from [2] to get anλ-independent estimate.

3. CONNECTION WITHBOBIENSKI´ ’S RESULT

In [1], the author considers a polynomial one-formω=MdH

H having a Darboux first integral:

H = (x−)^aP

k

Y

i=1

P_i^ai, M = (x−)P

k

Y

i=1

P_i (integrating factor),

3

whereP, P_i ∈R[x, y], a, a_i ∈R⁺andis a sufficiently small parameter. He imposes the following genericity assumptions

(1) The Darboux first integralHis regular at infinity.

(2) For = 0, the polycycleγ(0,0)consists of the edges γ_i(0,0)contained in a smooth part of the level curve P_i⁻¹(0). Any vertex p_ij, except (0,0), corresponds to the transversal inetrsection of level curvesP_i⁻¹(0)andP_j⁻¹(0).

(3) The polynomialP has a critical point of Morse type(0,0), i.e. P(x, y) =y²−x²+h.o.t.

Other polynomialsP_j satisfyP_j(0,0)6= 0, j = 1, . . . , k.

He considers a small polynomial perturbationω =M^dH_H

ω_,ε=ω+εη, η =Rdx+Sdy, R, S ∈R[x, y].

The linear part in perturbation parameterεof the Poincar´e first return map is given by the pseudo- Abelian integrals

I(, h) = Z

γ(,h)

η

M, γ(, h)⊂H⁻¹(h).

Under the above genericity assumptions, Bobie´nski proves the existence of a local upper bound for the number of zeros of corresponding pseudo-Abelian integrals I(, h). However, his study does not give the existence of a uniform bound in afullneighborhood of the parameter space(, h). The precise sense is clear after blow-up. Bobie´nski’s argument works well when one is studying the integral along cycles contained in a neighborhood of the big polycycleδ=δ₀₁tδ₀₂tδ₀₃tδ−t δ+tδ3t · · · tδk, see Figure 2. It corresponds to a sector in the(, h)-space. However, in order to have a study in a full neighborhood of (0,0) in the product space(, h), one must study also cycles landing on the exceptional level on a polycycle made of the same polycyle in the equatorial plane, but completed by any one of the continuous family of curvest = ^a+2_h joining two saddlesp₊and p−, see Figure 2. The study of the cycles in this sector is not done in Bobie´nski’s paper.

To complete the proof, which is the main result of this present paper, we assume that the pertur- bative one-formη vanishes to the order≥4at(x, y) = (0,0)(technical condition), but our study covers a full neighborhood of the polycycle in the product space with space of parameters. Also, in this paper Darboux first integral is more general, i.e.1 6=2. We prove the main result in full gen- erality by blowing-up the whole familyF_λand subsequently carefully piece-wise investigating the result combining Petrov’s technique [10], general Gabrielov type results and topological methods.

4. RECTIFYING OF DARBOUX FIRST INTEGRAL

Let us a establish a local normal form near the triple point (0,0,0) for the unfolding of the degenerate polycycleP₀....P_k = 0.

Proposition 4.1. Under the above assumptions A₂,A₃, there exists a local analytic coordinate system(x, y, λ)at(0,0,0)such thatHλ takes the form

H_λ = (x−λ)(y−x)⁺(y+x)⁻∆, (3) where∆is an analytic unity function∆(0,0,0)6= 0.

Proof. There exists an analytic coordinate system(x, y)at(0,0)such thatP1(x, y) =y∆1, P2(x, y) = x∆2 and Qk

i=3P_iⁱ = ∆3, where ∆1,∆2,∆3 are unities. In these coordinate and by Weier- strass preparation theorem we have P_λ = (x −f(y, λ))∆₄ where ∆₄ is a unit, ^∂f_∂λ(0,0) 6= 0

4

and ^∂f_∂y(0,0) 6= 0. A second application of Weierstrass preparation theorem allows us to write f(y, λ) = (y+g₀(λ))∆₅, where∆₅is a unity function and ^∂g_∂λ⁰(0)6= 0. Now we putx˜= _∆^x

5. Then

˜

x∆₅+ (y+g₀(λ))∆₅ = (˜x−y−g₀(λ))∆₅.

Finally, P_λ = (˜x−y−λ)∆˜ ₄∆₅, where λ˜ = g₀(λ). The normal form (3) can be obtained by a

linear rotation on(˜x, y).

5. DARBOUX INTEGRABLE FOLIATION

Consider two Darboux integrable foliations of dimension two in dimension three space with coordinates(x, y, λ)

F₁ ={M_λdH_λ

H_λ = 0}, F₂ ={dλ= 0}.

LetF = {M_λ^dH_H^λ

λ ∧dλ = 0}be the foliation of dimension one in a space of dimension three with coordinates (x, y, λ) which is given by the intersection of the leaves of F1 and F2. This foliation has a non-elementary singularity at the origin(0,0,0)which is the triple point. To reduce this singularity, we make the blowing-up of the triple point of the family in the product space (x, y, λ)of phase and parameter spaces. The family blowing-ups were introduced by Denkowska and Roussarie in [5].

5.1. Blowing-up the triple point. The blowing up σwith center at the origin (the triple point) is the projection ontoC³of a spaceW that is obtained by replacing the origin by the projective space CP²of all lines throught the origin:

W ={(p, q)∈C³×CP² :p∈q}

andσ :W → C³is defined by σ(p, q) = p. Outside the origin, a pointpbelongs to a unique line q, butσ⁻¹(0) = CP² which is called the exceptional divisor. If we write pin terms of the affine ccordinatesp = (p₁, p₂, p₃), andqin the corresponding homogeneous coordinatesq = [q₁, q₂, q₃], then the relationp∈qtranslates into the system of equationsp_iq_j =p_jq_i, fori, j = 1,2,3.

The projective spaceCP²is covered by three canonical charts: W₁ ={x6= 0}with coordinates (Y₁, E₁),W₂ ={y6= 0}with coordinates(X₂, E₂)andW₃ ={λ 6= 0}with coordinates(X₃, Y₃).

W₁, W₂ and W₃ define canonical charts on W, with coordinates (X₁, Y₁, E₁),(X₂, Y₂, E₂) and (X₃, Y₃, E₃)respectively. In these coordinates,σis given by the formulas:

σ₁ =σ|_W1 :x=X₁ y=X₁Y₁ λ=E₁X₁, (4) σ₂ =σ|_W2 :x=X₂Y₂ y=Y₂ λ=E₂Y₂, (5) σ3 =σ|W3 :x=X3E3 y=Y3E3 λ=E3. (6) We apply the blowing-upσto the one-dimensional foliation F on the three-dimensional space with coordinates(x, y, λ)given by intersection ofM_λ^dH_H^λ

λ = 0anddλ = 0. Letσ⁻¹F be the lift of the foliationF to the complement of the exceptional divisorCP².

Proposition 5.1. This foliationσ⁻¹F extends in a unique way to a holomorphic singular foliation σ^∗F onW which we call the blow-up of the original dimension-one foliationFby the mapσ. The foliationσ^∗F is regular outside of the preimage of the hypersurface{H(x, y, λ) = 0, λ= 0}.

5

5.1.1. Singularities of σ^∗F. The strict transform of the period annulus Dlies completely within the chart W₁. Then, we concentrate our study uniquely on this chart. Let σ^∗₁F be the restriction of the blown-up foliation σ^∗F to the chart W₁. We have σ is a biholomorphism outside CP² (exceptional divisor), all singularities of σ₁^∗F on W1 \ {X1 = 0} correspond to singularities of F. Thus, it suffices to compute the singularities ofσ₁^∗F on the exceptional divisor {X1 = 0}.

Explicitly, near the exceptional divisor{X₁ = 0}, the blown-up foliationσ^∗₁F is given by two first integralsλ=X₁E₁and ^λ_h^a =G, where

G=E₁^a(1−E1)⁻(Y1−1)⁻⁺(Y1+ 1)⁻⁻∆˜⁻¹,

∆˜ is unit of the form∆ =˜ c+X₁f, f is a holomorphic function anda =+₊+−. Consider the two-dimensional squareQ⊂CP² with verticesp+, p−, q+andq−. All levels curves{G= ^λ_h^a} insideQcorrespond to values of ^λ_h^a ∈[0,+∞].

Proposition 5.2. The singularities ofσ₁^∗F on the exceptional divisorCP² are located at the points p₊ = (0,1,0), p− = (0,−1,0), q₊ = (0,1,1)andq− = (0,−1,1). All these singular points are linearisable saddles, with eigenvaluesµ₊ = (₊,−a,−−), µ− = (−−, a, −), ν₊ = (0,−, ₊) andν− = (0,−, −)respectively.

Proof. Let us compute the eigenvalues at p₊, p−, q₊ and q−. Near p₊ the foliation σ₁^∗F is given by the two first integralsh =X^aY⁺ andλ = XE(we make a convenient variables change). By Hartman-Grobman theorem the vector field generating the foliationσ^∗₁F is given by

X(X, Y, E) = µ⁺₁X ∂

∂X +µ⁺₂Y ∂

∂Y +µ⁺₃E ∂

∂E,

such that the vector<(µ⁺₁, µ⁺₂, µ⁺₃),(a, ₊,0)>= 0and<(µ⁺₁, µ⁺₂, µ⁺₃),(1,0,1)>= 0. By short computation, we obtain

X(X, Y, E) = ₊X ∂

∂X −aY ∂

∂Y −₊E ∂

∂E.

Similar computation shows that there are local coordinates nearq₊ in which the vector field gen- erating the foliation is given by

Y(X, Y, E) = −Y ∂

∂Y +₊E ∂

∂E.

6. PROOF OF THEOREM 2.1

Let us fix some useful notations. Lett = ^λ_h^a, wherea =+−+₊andQ ⊂CP² is the two- dimensional square with verticesp₊, p₋, q₊andq₋. All levels curves{G=t}insideQcorrespond to values oft∈[0,+∞], where

G=E₁^a(1−E₁)⁻(Y₁−1)⁻⁺(Y₁+ 1)⁻⁻∆˜⁻¹.

6.1. Polycycles, relative cycles and normal form. The important advantage of making the blowing- up σ is to obtain a family of hyperbolic polycycles, i.e. at each intersection of two consecutive curves we have a saddle point. We consider the family of hyperbolic polycyclesδ

δ = σ⁻¹₁ (γ(0,0)\(0,0,0))∪(Q∩ {G=t})_R

, t∈[0,+∞], (7)

where(. . .)^Rdenotes the real part of a complex analytic set.

6

q+

q-

p+

p-

03

01

02

04

0

+

-

i

(

FIGURE 2. The foliationσ^∗₁F

6.1.1. Polycycles. Let δ₀ (edge) be the real part of the complex analytic set {X₁ = E₁ = 0}

joining two saddlesp− andp₊,δ₀₁ be the real part of the complex analytic set{X₁ = 0, Y₁ = 1}

joining the two saddlesp₊andq₊,δ₀₂be the real part of complex analytic set{X₁ = 0, Y₁ =−1}

joining two saddles p− and q−, δ₀₃ be the real part of complex analytic set {X₁ = E₁ = 0}

joining two saddles q− and q₊, δ₀₄ be the real part of complex analytic set {X₁ = 0, G = t}

joining two saddlesp−andp₊, δ₊ be the real part of the complex analytic set{Y₁ = 1, E₁ = 0}

joining two saddlesp1 andp3, δ−be the real part of the complex analytic set{Y1 = −1, E1 = 0}

joining the two saddles p− and pk and finally let δi be the real part of the complex analytic set {σ₁^∗Pi = 0, E1 = 0}, i= 3,· · · , kjoining the two saddlespmandpn-see Figure 2.

Let0≤m < M. Then, we can decompose the family of hyperbolic polycyclesδas follows:

(1) Ift∈[0, m[, we haveδ=δ0tδ−tδ+tδ3t · · · tδk.

7

(2) Ift∈[^m₂,2M], we haveδ =δ₀₄tδ₋tδ₊tδ₃t · · · tδ_k;

(3) Ift∈[M,+∞], we haveδ =δ01tδ02tδ03tδ−tδ+tδ3t · · · tδk.

6.1.2. Relative cycles. Letp₁,· · · , p_kbe the saddles points of the foliation σ^∗₁F which lie on the polycycle δ. Let δ⁰ = δ \ {p1,· · · , pk}. Choose a family of analytic transversals (of complex dimension two)Σ_x, through each point pointxinδ⁰. We can define a relative cycleγ (the part of the cycleδ(λ, t)) by a initial condition (starting pointx) and a end pointy =γ∩Σ_ywhich is going fromΣ_x. Concretly, we consider the relative cyclesδ₀(λ, t)corresponding to the edgeδ₀,δ₀₁(λ, t) corresponding to the edgeδ₀₁, δ₀₂(λ, t)corresponding to the edgeδ₀₂, δ₀₃(λ, t)corresponding to the edge δ₀₃, δ₀₄(λ, t) corresponding to the edge δ₀₄, δ₊(λ, t) corresponding to the edge δ₊, the relative cycleδ−(λ, t)corresponding to the edge δ− and the relative cyclesδ_i(λ, t)corresponding to the edgeδ_i, i= 3,· · · , k- see Figure 3.

q+

q-

p+

p-

03(

01(

0 (

+ (

- (

i (

FIGURE 3. The relative cycles

8

6.1.3. Normal form coordinates near the polycycles. Now we obtain normal forms in neighbor- hoods of each edge of polycyles

Proposition 6.1. (1) There exist a local chart(U₀,(X, Y, E))⊂W defined in a neighborhood ofδ₀ such that the blown-up foliationσ₁^∗F is given by two first integrals

λ=XE, t = (Y −1)⁻⁺(Y + 1)⁻⁺E^a.

(2) There exist a local chart(U₀₁,(X, Y, E))⊂W defined in a neighborhood ofδ₀₁such that the blown-up foliationσ₁^∗F is given by two first integrals

λ =XE, t=E^a(1−E)⁻Y⁻⁺.

(3) There exist a local chart(U₀₂,(X, Y, E))⊂W defined in a neighborhood ofδ₀₂such that the blown-up foliationσ₁^∗F is given by two first integrals

λ=XE, t=E^a(1−E)⁻Y⁻⁻.

(4) There exist a local chart(U₀₃,(X, Y, E))⊂W defined in a neighborhood ofδ₀₃such that the blown-up foliationσ₁^∗F is given by two first integrals

λ=X, t=E⁻(Y −1)⁻⁺(Y + 1)⁻⁻.

(5) There exist a local chart(U₊,(X, Y, E)) ⊂W defined in a neighborhood ofδ₊ such that the blown-up foliationσ₁^∗F is given by two first integrals

λ=XE, t=Y⁺(1 +X)ⁱE^a.

(6) There exist a local chart(U₋,(X, Y, E)) ⊂W defined in a neighborhood ofδ₋ such that the blown-up foliationσ₁^∗F is given by two first integrals

λ=XE, t=Y⁻(1 +X)ⁱE^a.

(7) There exist a local chart(Ui,(X, Y, E))⊂W, i = 3,· · · , k, defined in a neighborhood of δ_i such that the blown-up foliationσ₁^∗F is given by two first integrals

λ =E, t=Yⁱ(1−X)ⁱ⁺¹Xⁱ⁻¹ =t.

6.2. Proof of Theorem 2.1. Let δ be a polycycle. Let δ(λ, t) = σ⁻¹(γ(λ, h)) ⊂ W (dashed cycle, see Figure 2) be the pull-back of the cycle γ(λ, h) by the blowing-up map and δ be its corresponding polycycle. We define the integral

J(λ, t) = Z

δ(λ,t)

σ₁^∗ η

M_λ. (8)

This integral is considered as the pull-back of the pseudo-abelian integralsI(λ, h)by the blowing- upσ₁, i.e. J(λ, t) =σ₁^∗I(λ, t). The proof of Theorem 2.1 is reduced to the proof of the following theorem

Theorem 6.2. Letε > 0be sufficiently small. Then, for all|λ| < εthe number#{t ∈ [0,+∞] : J(λ, t) = 0}is locally bounded.

9

6.2.1. Variation operator. Firstly, let us recall some definitions, notation and general results. They will be useful later.

Definition 6.3. Given any multivalued function J defined in a neighborhood of the origin in C i.e. a holomorphic function defined on the universal coveringCf^∗ of C^∗. We define the rescaled monodromy as

Mon_(t,α)J(t) = J(te^iπα).

The variation is given as the difference between the counterclockwise and clockwise continuation Var_(t,α)J(t) =Mon_(t,α)J(t)− Mon(t,−α)J(t)

=J(te^iπα)−J(te^−iπα).

Definition 6.4. Let J be a multivalued function in two variables λ and t defined in universal coveringC²\ {λt^= 0}ofC²\ {λt= 0}. We define the mixed variation as

Var_(λ,β)◦ Var_(t,α)J(λ, t) =Var_(λ,β)(J(λ, te^iπα)−J(λ, te^−iπα)) =

J(λe^iπβ, te^iπα)−J(λe^−iπβ, te^iπα)−J(λe^iπβ, te^−iπα) +J(λe^−iπβ, te^−iπα).

Lemma 6.5. The variationsVar_(λ,β) andVar_(t,α)commute

Var_(λ,β)◦ Var_(t,α) =Var_(t,α)◦ Var_(λ,β).

Proof. The proof is a consequence of the monodromy theorem which says that: Ifγ₁, γ₂ are ho- motopic paths inC²\ {λt= 0}, thenψ_γ1 =ψ_γ2 whereψ_γ1 =Mon_γ1ψ andψ_γ2 =Mon_γ2ψ. We consider

γ₁(θ, φ) = (λ(θ, φ), t(θ, φ)) = λ, te^iθ

θ∈[0,α]t λe^iφ, te^iα

φ∈[0,β], γ₂(θ, φ) = (λ(θ, φ), t(θ, φ)) = λe^iφ, t

φ∈[0,β]t λe^iβ, te^iθ

θ∈[0,α].

The paths γ₁ andγ₂ are homotopic and this implies that ψ(λe^iαπ, te^iβπ)can be defined either as ψ_γ1 orψ_γ2. The same argument holds for the other germsψ(λe^−iαπ, te^iβπ),

ψ(λe^iαπ, te^−iβπ)andψ(λe^−iαπ, te^−iβπ).

6.2.2. Analytic properties. The integralJ(λ, t)has an analytic extension to the complex argument t(respλ). This is a multivalued function with unique branch pointt = 0(respλ = 0). As in [2], the key of the proof of Theorem 6.2 is the following.

Proposition 6.6. The integralJ(λ, t)satisfies the following iterated rescaled variations equation Var(t,α1)◦. . .◦ Var(t,α_k)J(λ, t) = 0. (9) whereα_iare polynomials functions in, ₊, −, ₃,· · · , _k.

Proof. Let us fixλ. We choose a hyperbolic polycycle δof the family (6.1). As in [2], using the different charts of Proposition 6.1 and partition of unity multiplying the blown-up one formσ₁^∗_M^η

λ

we can consider semilocal problem with a relative cycleδ_i(λ, t)(part of cycleδ(λ, t)) close to one edge (i-th edge) of the polycycle. Letδ_i^Cbe the complexification of the reali-th edge joining the singular pointspi−1, pi+1(saddles).

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(1) Ifα_i−1 6=α_i+1 (generic case), we have

Var_(t,αi−1)◦ Var_(t,αi+1)δ_i(λ, t) =[γ_i−1^, γ_i+1], (10) where[γ_i−1^, γ_i+1]is a complex (closed) cycle obtained as a lift of the commutator[γ_i−1, γ_i+1], whereγi−1 and γ_i+1 are paths in δ_i^C\ {pi−1, p_i+1} turning once counterclockwise around pi−1 andpi+1. This lifting[γi−1^, γi+1]vanishes by making third variation, i.e.

Var_(t,αi)[γi−1^, γ_i+1]≡0. (11) (2) Ifα_i−1 =α_i+1 (resonant case), we have

Var_(t,αi−1)δ_i(λ, t) =γi−1^γ_i+1⁻¹, (12) where γ^i−1γ_i+1⁻¹ is a complex (closed) cycle obtained as a lift of the figure eight loop γi−1γ_i+1⁻¹. This liftingγ^i−1γ_i+1⁻¹ vanishes by making second variation, i.e.

Var_(t,αi)γ^_i−1γ_i+1⁻¹ ≡0. (13) Finally, the variations commute so

Var_(t,α1)◦. . .◦ Var_(t,αk)δ(λ, t) = 0. (14) This argument is independent of the choice of polycycle, i.e. it holds for any hyperbolic polycycle

δof family (7).

Proposition 6.7. The rescaled variation with respect to λof the integral J(λ, t) is an integral of the formσ^∗_1M^η

λ along the figure eight loop

Var_(λ,1)J(λ, t) = Z

eight loop

σ^∗₁ η

M_λ. (15)

Proof. In the local chart (U₀,(X, Y, E)) of Proposition 6.1, the blown-up foliationσ₁^∗F is given by the two first integralsλ=XE andt=E^a(Y −1)⁻⁺(Y + 1)⁻⁻. Letγ₊andγ−be two paths inY^Cturning counterclokwise aroundp₊andp−which are parametrized by

ρ_± :θ∈[0,2π]7→







X(θ, λ, t)

Y(θ) = ±1 +εe^iθ E(θ, λ, t)

.

Then, we have

F±(λ, t) = Z

γ±

σ₁^∗ η M_λ =

Z 2π 0

ρ^∗_±σ^∗₁ η M_λdθ.

Also, we define two functions F1(λ, t) =

Z

`−

σ₁^∗ η

M_λ, F2(λ, t) = Z

`+

σ^∗₁ η M_λ,

where`−= [1−ε,−1 +ε]and`₊ = [−1 +ε,1−ε](segments). So, we obtain Var(λ,1)J(λ, t) = F−(λ, t) +F2(λ, t) +F+(λ, t) +F1(λ, t) =

Z

γ+`^−γ−`+

σ^∗₁ η M_λ,

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whereγ₊^`−γ−`₊is a closed path obtained as a lift of the pathγ₊`−γ−`₊which is contained inY^C

and homotopic to a figure eight loop.

Corollary 6.8. Near the ramification pointλ= 0, the functionJ(λ, t)admits the expansion J(λ, t) = J₁(λ, t) +J₂(λ, t) logλ, (16) whereJ₁(., t)is meromorphic andJ₂(λ, t) =Var_(λ,1)J(λ, t).

6.2.3. Proof of Theorem 6.2. The integralJ(λ, t) = R

δ(λ,t)σ^∗_1M^η

λ can be analytically continued to the universal coverC²\ {λt^= 0}ofC²\{λt= 0}. To estimate the number of zeros of the integral J(λ, t) =R

δ(λ,t)σ^∗_1M^η

λ we apply the argument principle.

Let us introduce some definitions which will be useful later.

Definition 6.9. Let f : Rⁿ× R → R. We shall say that f is a logarithmico-analytic function (LA-function) of type`in variableyif it has the following form

f(x, y) = F(f₁(x, y), . . . , f_m(x, y),logf_m+1(x, y), . . . , f_m+r(x, y)), whereF is a global sub-analytic function andfiare a LA-functions of type`−1iny.

Definition 6.10. A logarithmico-exponential function (LE-function) is a finite composition of global sub-analytic functions, exponentials and logarithms.

Let ∂Ω be the boundary of a complex domain Ωwhich consists of a big circular arc C_R1 = {|t| = R1,|argt| ≤ απ}, a two segmentsC^± = {r1 ≤ |t| ≤ R1,|argt| = ±απ}and the small circular arcCr1 ={|t|=r1,|argt| ≤απ}-see Figure 4.

The argument principle says that

#{t ∈Ω :J(λ, t) = 0} ≤ 1

2π∆ arg_∂ΩJ = 1

2π(∆ arg_CR

1 J+ ∆ arg_C±J+ ∆ arg_Cr

1 J).

(1) The boundedness of the increment of argument ∆ arg_C

R1 J. By Gabrielov’s theorem [6], the increment of the argument∆ arg_Cr

1 J is uniformly bounded.

(2) The boundedness of the increment of argument∆ arg_C^±J. Letα ∈ {α1,· · · , αk}. We use Schwartz’s principle

Im(J(λ, .))|_C^± =∓2iVar_(t,α)J(λ, t).

So ∆ arg_C±J ≤ #{t : Im(J(λ, .)) = 0} = #{t : Var_(t,α)J(λ, t) = 0}. Moreover, the variations commute so

Var_(t,α1)◦ · · · ◦ Var_(t,α)◦ · · · ◦ Var_(t,αk)J(λ, t) = Var_(t,α1)◦ · · · ◦ Var_(t,αk)(Var_(t,α)J(λ, t)) = 0.

Then, near the ramification point t = 0, the function Var_(t,α)J(λ, t) can be written as follows

Var_(t,α)J(λ, t) =F(e^αα1logt, . . . , e^{αkα logt},logλ) (17) whereF is a meromorphic function. The functionVar_(t,α)J(λ, t)is a LA-function of type 1 in variableλ. Then, by Lion-Rolin preparation theorem [8] this function has the following form

Var_(t,α)J(λ, t) =λ^q₀⁰λ^q₁¹G(t)U(t, λ0, λ1), U(0,0,0)6= 0 (18)

12

Re(t) Im(t)

0

C⁺

C^-

CR1

C_r1 𝛼𝜋

FIGURE 4. The contour∂Ω

withλ₀ =λ−θ₀(t), λ₁ = logλ₀ −θ₁(t), whereθ₀, θ₁,GandUare LE-functions. As the number of zeros of a LE-function is bounded, so #{t : Var_(t,α)J(λ, t) = 0}is uniformly bounded inλ.

(3) The boundedness of the increment of argument ∆ arg_Cr

1 J. Consider the following func- tional spaceP

P(m, M;α₁, . . . , α_k;λ) := {X X

c_jln(λ)t^αjnlogⁿt, c_jln ∈C, m≤α_jn≤M,0≤l≤k}.

Proposition 6.11. We haveJ2(λ, t) = Var(λ,1)J(λ, t) = O(λ^µ) uniformly int, for some constantµ >0.

Proof. Using the assumptionA₄, we haveσ₁^∗_M^η

λ isO(X₁), we conclude that, for all closed paths of finite lenght` < ∞contained in a sufficiently small neighborhood of the excep- tional divisor{X₁ = 0}. Since J₂(λ, t) =Var_(λ,1)J(λ, t)is the integrations ofσ₁^∗_M^η

λ over the lift of the eight figure on{X₁ = 0, G = t}, we conclude thatX₁ = O(λ)on this lift and

J₂(λ, t) =Var_(λ,1)J(λ, t) =λ^µt^ν(1 +. . .), µ >0.

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Lemma 6.12. The functionsJ₁(λ, .), J₂(λ, .)are two meromorphic families inλand satisfy the following rescaled variation equation with respect tot

Var_(t,α1)◦. . .◦ Var_(t,αk)J_i(λ, t) = 0, i= 1,2. (19) Then, there exists a family of meromorphic functionP₁(λ, .), P₂(λ, .)∈ P(...)such that

|J_i(λ, t)−P_i(λ, t)| ≤C|t|^M, uniformly in λ, i= 1,2

and J₂(λ, t)−P₂(λ, t) = O(λ^µ), µ > 0uniformly int and(J₂(λ, t)−P₂(λ, t)) logλ = O(λ^µlogλ). MoreoverJ(λ, t)6= 0. Then for sufficiently bigM:P₁(λ, t)+P₂(λ, t) logλ6=

0.

Proof. Using the linearity of the variation operatorVar, equations (9) and (16), we have Var_(t,α1)◦. . .◦ Var_(t,αk)J_i(λ, t) = 0, i= 1,2.

Lemma 4.8 from [2] yields that there exists an analytic (a priori meromorphic) families of functionsP_i(λ, .)∈ P(. . .)such that|J_i(λ, t)−P_i(λ, t)| ≤C|t|^M, uniformly inλ.

To estimate the limit of the increment of argument∆ arg_Cr

1J(λ, t)along the small cir- cular arc Cr1: limr1→0∆ arg_Cr

1 J, we investigate the leading term of J(λ, t) at t = 0.

By Lemma 6.12 we have J₁(λ, t) +J₂(λ, t) logλ−(P₁(λ, t) +P₂(λ, t) logλ)is O(t^M) uniformly inλ. For each element of parameters space, we can choose the leading termP of P₁(λ, t) +P₂(λ, t) logλ. By Gabrielov’s theorem, the increment of argument of P is bounded.

ACKNOWLEDGMENTS

It is pleasure to thank Pavao Mardeˇsi´c (Universit´e de Bourgogne-Dijon) and Daniel Panazzolo (Universit´e de Haute Alsace-Mulhouse) for their constants suport in this work.

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UNIVERSITE DE´ BOURGOGNE, INSTITUT DEMATHEMATIQUES DE´ BOURGOGNE, U.M.R. 5584DUC.N.R.S., B.P. 47870, 21078 DIJON CEDEX- FRANCE.

E-mail address:aymenbraghtha@yahoo.fr

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